Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional...
Transcript of Dimensional analysis - zeus.plmsc.psu.edumanias/MatSE447/23-26_FlowsInProcessing.pdfDimensional...
Dimensional analysisThe Buckingham π Theorem - If there are m physical quantities con-
taining r fundamental units, then there are π = m− r dimensionless groupsthat are independent.
Procedure: 1. Write down all relevant physical quantities and theirfundamental units.2. Determine π.3. Write down the π dimensionless groups.
Example: Flow through a circular pipe
dP
dx= f(Q, R, µ) m = 4
dP
dx
(dynes
cm3=
g
cm2s2
)Q (cm3/s)
R (cm)
µ
(dynes
cm2s =
g
cm s
)g, cm, s r = 3
∴ π = m− r = 1
1
µ
dP
dx
(1
cm s
)only way to get rid of grams
1
µQ
dP
dx
(1
cm4
)only way to get rid of seconds
R4
µQ
dP
dx(dimensionless)
Dimensional Analysis ⇒ dP
dx∼ µQ
R4
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Dimensional analysis
Dimensional Analysis ⇒ dP
dx∼ µQ
R4
Not as useful as the exact result:
∆P
L=
8
π
µQ
R4
because exact result has the prefactor.
How did we get so much information? We cheated!
Used “experience” to guess that ∆P and L would only enter as dP/dx =∆P/L, and that the velocity v would only enter though Q.
“Honest” Treatment:
∆P = f(Q,R,L, µ, v)
∆P (g/cm s2)
Q (cm3/s)
R (cm)
L (cm)
µ (g/cm s)
v (cm/s)
m = 6 r = 3 ∴ π = m− r = 3
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Dimensional analysisEXAMPLE: FLOW IN AN EXTRUDER
dP
dx= f(Q, Ds, hs, Ns, φs, µ)
( g
cm2s2
)Q (cm3/s) volumetric flow rate
Ds (cm) barrel diameter
hs (cm) flight depth
Ns (1/s) screw rotation rate
φs (dimensionless) helical angle of screw
µ (g/cm s) viscosity
m = 7, r = 3 ⇒ π = m− r = 4
Four dimensionless groups
φs,Ds
hs
,Q
NsD3s
1
µ
dP
dx
(1
cm s
)(cancels grams)
Ds
µNs
dP
dx(dimensionless pressure gradient)
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Dimensional analysisEXAMPLE: FLOW IN AN EXTRUDER
Ds
µNs
dP
dx= f
(φs, Ds/hs, Q/(NsD
3s)
)Optimum φs = 23◦ (universally utilized)Most extruders are geometrically similar and designed to provide a certain
pressure at the die:
∴Ds
hs
∼= constant
∴Ds
µNs
dP
dx∼= constant
If other three dimensionless groups are constant, then the last one mustalso be constant.
Expect Q ∼ NsD3s (not obvious)
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Melt Fiber-Spinning
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Melt vs. Solution Fiber-Spinning
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PIPE EXTRUSION: Die Swell is Crucial!
Mandrel Supports:
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Profile Extrusion
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Sheet ProcessingSHEET EXTRUSION
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Sheet ProcessingCLAM SHELLING IN A SHEET DIE
Figure 1: Pressure is very high (100atm) at the center of the die.
Figure 2: High pressure causes die to creep. After extensive extrusion, sheetwill be thicker in the center.
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Sheet ProcessingPOST-DIE SHEET FORMING
Calendering Thin Sheet:
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COEXTRUSION LAYER PACKS
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COEXTRUSIONBARRIER POLYMERS
EVOH (ethylene-vinyl alcohol) random copolymer with 30%EPVDC (polyvinylidene chloride) [CH2CCl2]
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COEXTRUSION
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COEXTRUSION
6
COEXTRUSION
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FILM BLOWING
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FILM BLOWING
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Profile
Extru
sion
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Wire Coating
Combination of Poiseuille flow (pressure driven) and Couette (drag) flow dueto the moving wire.
First we do the Couette Flow problem assuming ∆P = 0.
vr = vθ = 0 vz = vz(r)dP
dr=
dP
dθ=
dP
dz= 0
Continuity is identically satisfied
N.-S.: µ1
r
d
dr
(rdvz
dr
)= 0
d
dr
(rdvz
dr
)= 0 r
dvz
dr= C1
dvz
dr=
C1
rvz = C1 ln r + C2
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Sheet ProcessingTHERMOFORMING
R.G. Griskey, Polymer Process Engineering, chapter 10
J. Florian, Practical Thermoforming (Marcel Dekker, 1987)
Extrude a Sheet, Clamp it, Heat it up, and ...
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THERMOFORMINGVACUUM FORMING
∆P = 1atm.
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THERMOFORMINGVENTING
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THERMOFORMINGPRESSURE-ASSISTED
To exceed ∆P = 1 atm. use pressurized air.
Avoid trapped air by using vacuum.
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